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reference request – The Guinand-Weil categorical formulation for Hecke characters

The Guinand-Weil formulation for the Riemann zeta operate is

commence{aligned}&Phi (1)+Phi (0)-sum _{rho }Phi (rho )={}&sum _{p,m}{frac {log(p)}{p^{m/2}}}{Big (}F(log(p^{m}))+F(-log(p^{m})){Big )}-{frac {1}{2pi }}int _{-infty }^{infty }varphi (t)Psi (t),dtend{aligned}

the place $rho$ runs over the nontrivial zeros of the Riemann zeta operate, $p$ runs over the primes, $m$ runs over the constructive integers, $F$ is a quickly lowering operate, and $Phi(1/2+it)=varphi(t)$ is the Fourier remodel of $F$.

To quote Wikipedia:

More usually, the Riemann zeta operate and the L-series can breathe changed by the Dedekind zeta operate of an algebraic quantity bailiwick or a Hecke L-series. The sum over primes then will get changed by a sum over prime beliefs.

**Q**: What is the Guinand-Weil formulation for Hecke L-functions, the place there’s a sum over Hecke L-function zeros of the Fourier remodel of a quickly lowering operate on the LHS, and a summation over prime beliefs on the RHS?

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